Zad. 1
A = (-4, -1)
B = (1, 4)
C = (5, 2)
Rownanie prostej AB
[tex]\left \{ {{-1=-4a+b} \atop {4=a+b /*(-1)}} \right. \\+\left \{ {{-1=-4a+b} \atop {-4=-a-b}} \right. \\-1-4=-4a-a\\-5=-5a /:(-5)\\1=a\\-1=-4*1+b \\-1=-4+b /+4\\3=b\\y=x+3[/tex]
Rownanie prostej AC
[tex]\left \{ {{-1=-4a+b/*(-1)} \atop {2=5a+b}} \right. \\+\left \{ {{1=4a-b} \atop {2=5a+b}} \right. \\1+2=4a+5a\\3=9a /:9\\\frac39=a\\\frac13=a\\-1=-4*\frac13+b\\-1=-\frac43+b /+\frac43\\-\frac33+\frac43 = b\\\frac13=b\\y=\frac13x+\frac13[/tex]
Rownanie prostej BC
[tex]\left \{ {{4=a+b/*(-1)} \atop {2=5a+b}} \right. \\+\left \{ {{-4=-a-b} \atop {2=5a+b}} \right. \\-4+2=-a+5a\\-2=4a /:4\\-\frac24=a\\-\frac12=a\\4=-\frac12+b /+\frac12\\4\frac12=b\\\frac92=b\\y=-\frac12x+\frac92[/tex]
Zad. 2
a) P=(-2. 1), Q=(2, 5), C=(4, 8)
Rownanie prostej PQ:
[tex]\left \{ {{1=-2a+b /*(-1)} \atop {5=2a+b}} \right. \\+\left \{ {{-1=2a-b} \atop {5=2a+b}} \right. \\-1+5=2a+2a\\4=4a /:4\\1=a\\1=-2*1+b\\1=-2+b /+2\\3=b\\y=x+3[/tex]
Czy punkt C nalezy do prostej PQ?
[tex]8=4+3\\8\neq 7[/tex]
Punkt C nie nalezy do prostej PQ.
b) P=(-1, 7), Q=(3, -1), C=(-2, 9)
Rownanie prostej PQ :
[tex]\left \{ {{7=-a+b /*(-1)} \atop {-1=3a+b} \right. \\+\left \{ {{-7=a-b} \atop {-1=3a+b}} \right. \\-7-1=a+3a\\-8=4a /:4\\-2=a\\7=-(-2)+b\\7=2+b /-2\\5=b\\y=-2x+5[/tex]
Czy punkt C nalezy do prostej PQ?
[tex]9=-2*(-2)+5\\9=4+5\\9=9[/tex]
Punkt C nalezy do prostej PQ.
